# Events for 05/03/2022 from all calendars

## Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Daniel Perales, University of Waterloo

Title: On the anti-commutator of two free random variables

Abstract: Let a, b be freely independent random variables in a non-commutative probability space. Based on some considerations on bipartite graphs, we provide a formula to express the n-th free cumulant of the anticommutator ab+ba as a sum indexed by subset Y_{2n} of non-crossing partitions of {1,2,...,2n}. Specifically, Y_{2n} is the set of partitions that separate odd elements and where blocks with only even elements have even size. We will study the sets Y_{2n} and obtain new results regarding the distribution of ab+ba. For instance, the size |Y_{2n}| is closely related to the case when a,b have a Marchenko-Pastur (free Poisson) distribution of parameter 1. We will also provide a formula with the sum indexed by cacti graphs, and discuss a natural generalization to study quadratic forms in $k$ free random variables. The talk is based on the preprint arXiv:2101.09444.